calculation of Effective area covered

i am new to this forum, i have a problem with calculating the effective area. Please see the attached doc file for figure information. In the figures, circles arranged in triangular,square and polygon shapes.

Problem: how to calculate the effective area covered by the circle ( you can ignore the boundary circles that means consider only the shaded region). Now if you increase the number of circles on the each side( for triangle and square) then the number of circles inside shaded area increases.

can any body provide the proof for calculating the effective area or point out the links

To me it looks like it is already proved by somebody but i couldn't find out on the internet.

you can think in this way: a circle with radius "r", arranged them in different topologies ( triangular, square).. As shown in figure, for triangular topology, each side has 5 circles and the circles in the figure are intersect in a way that leaves no uncovered space...

Effective area covered by a circle in the shaded region is , area covered by a circle with out considering the intersection area formed with other neighbor circles..